Understanding Determinants and Their Applications

Hello students welcome to your channel GodGlasses GodGlasses means mess and fun In today's class we will be doing one shot determinant Determinant is a very important topic By combining matrices and determinant we get a unit of 10 marks I have already done one shot of matrices I am giving the link of the playlist above You can see all the one shots there which are available till now And which will be available in the future Before moving forward, I would like to tell you that for your board exam preparation, the God Classes question bank and sample paper are available on both the God Classes apps. So you can order soft copy and hard copy there as you are comfortable and you can do the preparation of your board exam very well. And one more thing I would like to say that if you are liking the videos, then you must subscribe to our channel, press the bell icon and share the videos with your friends so that they can also take full benefit of these videos. So here the first concept is to see what is determinant Determinant is a number, a real number which is associated with any square matrix So here we have to remember that determinant always comes from square matrix To every square matrix A we can associate a unique real number called determinant you can write the determinant like this or you can write it as a positive or negative value when we write the determinant, we have one element in one matrix we have a matrix and we have written 2 here so if we calculate the determinant of this then it remains the same there is no change in this if you have a matrix of 2 by 2 1,3,2,5 so if we calculate the determinant of this so for this we have to multiply 1 by 5 minus 3 into 2 so it will be 5 minus 6 means minus 1 if you take out the determinant of 3x3 matrix 1,3,0,2,1,4,1,3,2 now when you take out the determinant first we take this element 1 then we will leave this element in the row and column ok we will leave this element and this element now we will solve the remaining element so after leaving these two, we will have 2-12 so here we will write 2-12 which means we will get minus 10 so after leaving this, we have to solve this so what will we have left? we will get minus 10 after that when we take this element then we will write its sign by changing it minus 3 now we will leave this row and this column so 2x2 is 4 then minus sign in the middle and 4 from here 4-4 is 0 here it is 0 so if you write 0 in it, it will not matter its answer will remain 0 so from here you have minus 10 this will be 0, it is 0 so its determinant is minus 10 you can remove the determinant from any row but whenever you remove the determinant remember that whenever you take this element For example, we have taken 3 here, so we have to change its sign If it is minus, we will change it to plus, if it is plus, we will change it to minus Similarly, if you write this element, then you have to write this by changing it to minus This too and this too These four elements, you have to write them by changing them to minus Whether you take it out of row or column, no matter how you take out the determinant, the answer will be the same For example, if we want to take it out of first row, then what will be done by first column answer will be same everywhere first we write 1 then this element will be minus 2 now we will leave this row and this column this will be 6 minus 0 6 is done then plus 1 this element is written now we have to leave this row and this column 3 4 is 12 minus 0 12 is done So this becomes minus 10, minus 12, plus 12, cancel and the answer is minus 10. So you can start from anywhere, the answer has to be the same. Now after that, let's remove the minor. So what is a minor? A minor is also a determinant of order n. n should be greater than or equal to 2. The value of n should be greater than or equal to 2. That means a minor of 1 by 1 should not be taken. To take a minor, at least a 2 by 2 order matrix should be there. Now how will we take a minor? Then the determinant of order n minus 1 obtained from the determinant after deleting its ith row and jth column is called the minor of the element. So, it has become a bit theoretical, let's see how to do it in a bit practical way. Suppose you have A matrix. 1, 2, 3, 4. What you have to do is to find the minor. So, first you have to find the minor of this element. So, we denote the minor as m11. That is, find the minor of the first row and first column. So, it is saying that the row and column of the element you want to find the minor of, delete it. That is, this row. and this column is deleted here you have 4 left this row and this column are deleted 4 left, so this is your answer this is how you will find the minor first row second column first row second column this delete you have 3 left second row first column second row first column you have 2 left second row second column you have 1 left so these are their minors Now we have to find the cofactor. The value of the minor you have found will remain the same. We denote the cofactor with capital A or C. I will write it with capital C. The value of the cofactor will remain the same. The only difference is the sum of i and j is odd. ij means the number written in the base as you can see here 1 and 2 have odd sum so you have to change the sign if it is plus then you have to minus and if it is minus then you have to plus as you can see here the sum is also odd so you have to change the sign this is the difference so whether you are taking the minus of 2 by 2 or 3 by 3 the co-factor you will take in all of them, you have to change the sign of them where the sum of i and j is odd number will come now we will take out 3x3 cofactors and we will do that in the next questions here I took 2x2 to explain the concept now it is said that we have already done this that if you take out any determinant then you expand any row or column so that your determinant remains the same now you have two determinants so now we have determined the value of x so here it is not like a matrix that we have taken the corresponding element and taken x as 4 equals to 6 so we have got the value of x as 4 now we have to find the determinant 3 into 0 is 0 then minus sign minus 24 from here we have minus 3 then minus sign and here we have x cube so this is plus 24 take this minus 3 here and it will be 27 you can write 27 as minus 3 cube and write it as minus x cube if you compare both then you will get x value x value will be 3 minus 3 should be there here it is plus 27 so we write it as this and we write it as this so we get x value minus 3 ok here it says find the minor of the element 7 in the determinant this now here 3 by 3 you have it is asking for the minor of element 7 where is element 7? here it is it is asking for its minor so which place is it? 2nd row 3rd column so 2nd row 3rd column minor means delete 2nd row and delete 3rd column now the remaining you have find its determinant minus 4 then minus sign 3 into 1 is 3. Then it will be minus 7. Now if you were asked about the cofactor of this, if you were asked about the cofactor of this, then we would write 7. Why? Because look at its sum. 2 over 3. 5 is its sum. Odd is its sum. So whatever sign had to come here, you have to reverse that sign. If it is minus, then you have to do plus. If it is plus, then you have to do minus. Find the minor and cofactor of A and check whether this. So here you have to remove all the minor and cofactors So we have said to remove the minor M11 M12, M13 then M21, M22, M23 M31, M32, M33 Solve this M11 first row, first column will be left determine this, 4 will be 0 then minus sign in the middle pi 4 is 20 so this is minus 20 now first row we will leave second column 6 7 is 42 then minus sign 4 1 is 4 so this is minus 46 first row third column 30 minus 0 so this will be 30 after that come to second row 2nd row 1st column so 2nd row and 1st column means minus 3 and minus 7 21 5 5 goes to 25 means it becomes minus 4 2nd row 2nd column minus 14 and minus 5 2nd row 3rd column 10 from here it is minus 3 so it becomes plus 13 third row first column third row first column minus 12 minus 5 into 0 third row second column third row second column 8 minus the third row second column 8 minus 30 so I'll take a minus 22 third row third row third column 0 minus minus 18 so this is plus 18 so this is the minus now you can calculate the co-factor what will be the co-factor let's denote it C11, C12, C13 C21, C22 C23 C31 C32 and C33 So C11 means You have this value of minus 20 Which sign you have to change in cofactor Which has some odd So what is the sum odd See this is the sum odd This is the sum odd This is the sum odd This is the sum odd You have to change the sign of these values Here minus 46 is coming Here we will write plus 46 Here 30 is coming here minus 4 is coming, plus 4 here minus 19 so this will remain the same here 13 minus 13 here minus 12 this will be plus 12 here minus 22 this will remain the same, minus we have to change the sign, so this will be plus 22 this will be 18 Now it is saying that the cofactor is gone Now it is asking us to verify this property So this property says that if we take any row element And if we multiply it with any other cofactor Then the answer will always be 0 If we multiply the row we are talking about with the same cofactor Then the answer always comes equal to the determinant But if we take any other row then the answer will always be 0 See how we are verifying here a is a cofactor we have written c here and given a there so here a11 a11 means 2 into a31 a31 means c31 c31 is minus 12 plus a12 first row second column means minus 3 into a32 32 is 22 plus a13 which is 5 into a33 which is 18 so from here it will be minus 24 minus 66 plus 90 if we add these two minus 90 plus 90 the answer is 0 so we have verified this property Next, you need to remember the properties of determinant These are the properties that will help you solve the questions These properties will be useful in MCQ questions Here, you don't have to do the questions like prove that You will use these properties in the rest of the questions In the first property, if each element in a row or column of a determinant is 0 Then the value of determinant is 0 In any determinant, if you want to solve the question All the elements are given zero in row or column So write anything here It doesn't matter anything, the answer will always be zero If each element on one side of the principal diagonal of a determinant is zero Then the value of determinant is the product of the diagonal element This means that if you have a diagonal element on one side of the diagonal element If the elements on one side of it are given zero The determinant of this will be the answer of the diagonal element The multiplication of these 6 Now it can be 0 here or here It can be 1,2,3 So 0 will be here And something else here Then also the answer will be 0 And both sides will be 0 sorry, the answer is 6 the answer is 6 the diagonal element will be multiplied whether it is this situation or that situation the determinant will remain the same the value of determinant remains unchanged if its row and column are interchanged if you interchange row and column then there will be no difference in the determinant interchanged row and column means that you are transposing it so it is saying that if you are going to use matrix or the transpose of the determinant, then it will remain the same. This property means this. If any two row or column of a determinant are interchanged, then the value of determinant changes by minus sign only. In the previous property, we were changing the row into column. We were making the first row into first column, i.e. we were doing transpose. But here it is saying that if you have a matrix, you are taking out the determinant a, b, c. D E F G H I Let's assume that we have a delta here Now I will interchange one row with another row I will make the first row as the second row and the second row as the first row So the answer will be minus sign So it will be minus delta If each element of any row or column of a determinant are identical or proportional, then the value of determinant is zero. If any determinant is 2a, 2b, 2c, then the answer is zero. Because these two rows are proportional. So either it is proportional or identical. so if here is a, b, c then also the answer will be zero and here if 2a, 2b, 2c means if you find the ratio of these two then also the answer will be zero if each element of any row or column of a determinant is multiplied by the same number k then the value of the new determinant is k times the value of the original determinant so here it says that if you have this determinant The answer is delta. If I multiply one row by K, K A, K B and K C, the answer is K multiplied in the determinant. If we multiply one row, it will be multiplied once. If you multiply the second row, then k will be square if you multiply third row also then k will be cube so from here a property is created if you calculate the determinant of k into a k into a means we have multiplied all the elements with k and the order of this will be n by n then k will be multiplied here k will be multiplied by 3 times in the 3x3 matrix k will be multiplied by 3 times in the original determinant so this property will be multiplied by the order of k Next, the sum of the product of any row or column with the cofactor of the corresponding element of the sum other row for the column is 0 Here it is said that if you multiply any row or column element with its cofactor If you multiply by the cofactor of another row, the answer will always be 0 which we just proved We took the element of first row and the cofactor of third row We multiplied them and added them together So the answer is 0 So this is the same property If A and B are square matrices of the same order Then determinant of AB equals to determinant of A into determinant of B If you determine AB, you can write it like this Determinant of A into Determinant of b Now using this property we get a result That if you want to get the determinant of a to the power n Then you simply have to Get the determinant and do it n times So what will be the benefit of this? Suppose you want to get the determinant of a to the power 4 So you don't have to Multiply a by 4 You will get the determinant of a And do it to the power 4 So this will be easier Now the question is, a and b are square matrices of the order 3 such that determinant of a equals to minus 1, determinant of b equals to 3. Then find the value of this. So the properties we have done now, you have to use those properties. First of all, take out the scalar. Its order is 3 by 3. So in this, we are using the property of k into a determinant. So how many times k comes out of this is the order of this So here you are taking out 3 How many times will you take out? 3 times So 3 power 3 into determinant of AB Now here you put the property that determinant of AB is equal to this So 27 into determinant of A is minus 1 and this is 3 So it will be minus 81 If A is a square matrix and determinant of A equals to 2, then write the value of this when A is the transpose of matrix A. You have to find its determinant. So, you can apply this property here that we will do determinant of A and determinant of A transpose separately. So, the result remains the same. Now, we have also read a property that whether you find A's determinant or its transpose's determinant, both are equal. So, I can write the determinant of A here So, this will be the square of the determinant of A A is the determinant of 2 2 power 2 means the answer is 4 Now, the concept is Area of Triangle Area of Triangle is the area of the three vertex of a triangle The formula of area of triangle is The sign here is the sign of determinant not of matrix Determinant you have to find is x1,y1,x2,y2,x3,y3 Keep this and in the last column always keep 1,1,1 If you solve this then this will be your area of triangle If your area of triangle is negative because area is never negative then we can neglect the minus sign If you find the area of triangle is minus 10, then the final answer is area of triangle equals to 10 square units. If you find the area of triangle is zero, then the points are collinear points. Because when the points are in the same line, then the triangle will never be formed. If the triangle is not formed, then the area of triangle will not be formed. So the area of triangle will be equal to zero. Now let's find the area of triangle So what was the formula? 1 by 2 In the formula write 5, 4, 2, minus 6, minus 2, 4, 1, 1, 1 Simplify this First we took 5, minus 6 and from here minus 4 then we will change the sign this one and this one this is minus 2 and the sign in between is minus plus 2 this one and this one 8 minus 12 this will be minus 10 10, 5 is 50 4 is 16 minus 4 So, everything is in minus, so see how much is here, here 20, 50, 70 is here. So, 1 into minus 70 means minus 35. So, when you write the area of this, you will not write minus 35. You will write the area of triangle is 35 square unit. For what value of k the point are collinear? Now tell me the value of k to make this point collinear Collinear means that the area of triangle is equal to zero because the point is collinear If you take the area of triangle equal to zero, then it will be half From here, 3 minus 2, 8, 8, k, 2, 1, 1, 1 So this half will go to zero, its work is finished With 3, 8 minus 2 6 is done. Then we will write by changing its sign. We will leave this and this. 8 minus k plus 1. We will leave this and this. 16 minus 8k. From here 18 will come. From here 2 is 16 minus 2k. From here 16 minus 8k. So minus 8 minus 2 minus 10k is done. So, we will take 10k there. 16 is 32 and 18. 50 equals to 10k. So, k will be 5. After that, the next question is in the next concept, we have adjoint. What does adjoint mean? Let a be the square matrix of order greater than or equal to 2. Means, the minimum order is 2. then adjoint of A is the transpose of cofactor if we write the cofactor transpose then it becomes adjoint of A and we will denote it as adjoint of A it has some properties of adjoint if you write adjoint of A then we write reverse law in transpose similarly you will write reverse law here adjoint of B adjoint of A To find the adjoint of A transpose, you can find the adjoint of A and then transpose it. Adjoint of adjoint of A. So, what will be its formula? Determinant of A power n minus 2 into A. Here, the determinant of adjoint of A. Its formula is determinant of A power n minus 1. If you add A with it, then it will be determinant of A power n. a into adjoint of a or adjoint of a into a so this will be equal to determinant of a into i so keep these properties in mind and use them in questions then comes singular matrix you can call a matrix singular only when determinant of a is 0 if determinant of a is not 0 then we will call it non singular matrix now it is saying that you have to tell the value of k if it is a singular matrix So, singular matrix means that its determinant is 0. So, we will get its determinant as minus 8 minus 3k equals to 0. So, 3k's value is minus 8. That means k's value will be minus 8 upon 3. Inverse of a matrix. A square matrix A of order n is invertible of their existing square matrix of the same order such that AB equals to BA equals to I. This means that if we do AB, or BA, which is equal to I In that case, we will say that A inverse is equal to B You can say that B is the inverse of A and you can also say that A is the inverse of B if AB and BA are equal to identity matrix Every invertible matrix possesses a unique inverse Whichever invertible matrix exists, you can find its inverse then A will be the inverse of A. It will not happen that two different inverses of one matrix will come. If A is an invertible matrix then inverse of A is this. If you want to get inverse of A, then it will be equal to this. We will denote inverse of A as such. A square matrix is invertible if and only if it is non-singular. If matrix is non-singular, then only its inverse will be derived, otherwise it will not be derived. If determinant of A is 0, then its inverse will not be derived. Because the formula to get inverse is adjoint of A divided by determinant of A. So it is not possible to get inverse. That is why determinant of A should always be non-zero. Means non-singular matrix should be formed. If A and B are two invertible matrix of the same order then A B is also invertible. If A and B are both invertible matrix, means we can get both inverse. So the matrix which will come even after multiplying A B will be invertible. And the inverse of A B will be equal to B inverse into A inverse. So here also reverse law is applied. So reverse law is applied in transpose also. It is applied in the adjoint and in the inverse as well. If A is an invertible matrix, then A transpose is also invertible. If A inverse is obtained, then A transpose inverse will also be obtained. And if you do inverse of A transpose or inverse of transpose, then both of them remain equal. If A is an invertible matrix, then determinant of A inverse. If you want to find the determinant of A inverse, then simply 1 upon determinant of A will be obtained. For what value of k the matrix has no inverse? He says that the matrix does not have an inverse. Inverse does not mean that the matrix is singular. If it has no inverse, then determinant of a equals to 0. So, we will get the determinant of 10 minus 3k equals to 0. So, the value of 3k is 10 and the value of k is 10 by 3. Okay. If the matrix is in non-singular, then find the value of x. If the matrix is non-singular, then non-singular means that the determinant of A is not equal to 0. So, find the determinant of A. 6, minus 2 and minus 10. Minus 12, minus x. We will leave this column as 4 plus 20 and this is 24 and we will leave this column as 10 minus 10 which is 0 0 is not equal to 0 So, this will be minus 72 and this will be minus 24x plus 0 is not equal to 0 Take 72 there So, the value of x will be not equal to 72 upon minus 24. That means, x should not be equal to minus 3. If a equals to this, then find x such that this. Then find a inverse. This means that, first of all, if you have this a, then you have to find the value of x, if we have this relation. So, according to this relation, a square xA is equal to xA-2i a square means 3-2, 4-2 equals to xA-2i if we want to square it, we have to write it again so this will be 9 minus 8, 1 is done. minus 6 plus 4, minus 2, 12 minus 8, 4, minus 8 plus 4, from here 3x minus 2x, 4x minus 2x, even get 2, 0, 0, 2. 3x-2, minus 2x, and 4x, minus 2x, plus 2. So, from here, since both the matrices are equal, keep this element equal to this. 3x-2 equals to 1. So, 3x equals to 3. So, the value of x is 1. But, is the value of x coming from each one or not? Do check this. So if you put 1 here, it will become minus 2. So here minus 2 is coming. If you put it here, it will become 4. 4 is coming. If you put it here, it will become... It will be zero but here it is minus 4. So we have done something wrong here. Okay, here it is minus 2. So here it will be minus 2. Now if you keep it here, it will be minus 4. So always check that if the value of x is different from two places, then it means that there is no value of x. So, we have got the value of x. The value of x is 1. Now, it is saying that hence find A inverse. Now, find its inverse as well. Now, how will we find the inverse? Whenever such a question arises, we will not directly apply the inverse formula that adjoint of A divided by determinant. We will use this equation. Now, we have got A square equals to x into A minus 2 into I. So, what we do is we multiply A inverse here as well. we multiply here as well and here as well if we multiply A inverse with this, then A will be saved if we multiply A with A inverse, then identity is saved and if we multiply identity with any matrix, then the same matrix is saved now inverse is here, take it this side and take it this side We have taken the value of x as 1. So, we will put 1 here. Let's put 1 first. Now, we will put 1. If we put 1 instead of x, then we will have i. We will take a here. Let me show you because x equals to 1. So, you have i is 1, 0, 0, 1 a is 3 minus 2 4 minus 2 So we have to find A inverse So we will take this 2 here 1 by 2 1 minus 3 minus 2 0 plus 2 0 minus 4 1 plus 2 Divide all these by 2 So minus 1, 1 minus 2 and 3 by 2 This will be your answer After this we have solution of a system of linear equation using inverse of matrix i.e. matrix method. There are chances of getting question of matrix method. In matrix method, we do solution and we just have to tell if the system is consistent or not. So how do we check consistency? If you have a non singular matrix, i.e. if we have determinant of A is not equal to zero then we will say that the unique solution of the system is and how to find the solution? x equals to a inverse b we will use this to solve the question and system is called consistent and we will say that system is consistent we can find a solution to this but if we have a singular matrix which means if determinant of a is equal to 0 then you have to calculate b in adjoint of a into if b in adjoint of a into is equal to 0 then we will say that system is consistent but there is no solution to it, there are infinite solutions to it but if b is not equal to 0 in adjoint of a into then we will say that there is no solution to the system and its solution is inconsistent, so its solution is not possible here it is telling you to solve it with inverse of coefficient matrix So here we have to use the help of the equation to make the matrix A So here we will take the coefficient of x and y Then we will make a matrix B We will write x here So in x we will take the variable we are using So we are using x and y You have to write this in the column And write the constant in the right hand side with B so we have 5 and minus 1 now the first thing we have to do is to find the determinant of A because if it is 0, then we have to check the solution whether it is possible or not so here we have done, it will be 3 then minus 2, which means the sign in the middle is minus, plus 2 which means 5 which means 0 is not equal to 0 0 is not equal to 0 means we can find the inverse of it To find the inverse, we need to find its adjoint We used to transpose the cofactor But here we have a 2x2 matrix If we find the adjoint of 2x2 matrix, we can find its direct In the 2x2 matrix, we need to change these two elements So, instead of 1, 3, we have 3, 1. And we have to change the sign of both of them. This will be 1 and this will be minus 2. So, you can always directly remove the adjoint of 2 by 2. But to remove 3 by 3, you have to remove its cofactor. Now, you have to remove its inverse from here. The inverse formula is adjoint of A divided by determinant of A. Now, what is the adjoint you have? 3, 1, minus 2, 1. we have to divide it by its determinant which is 5 so we will write 1 upon 5 this is the inverse now the relation we have made according to this we have if we multiply a by x then it is equal to b so we have to find the value of x so we will take a there which means a inverse into b now the value of a inverse is 1 upon 5 3x1-2x1 into b b is 5 minus 1 we will solve this 3 x 5 is 15 minus 3 15 minus 3 is 12 no 15 minus 1 is 14 this will be minus 10 and minus 1 minus 11 we will divide by 5 this will be 14 upon 5 minus 11 upon 5 So you will get the value of x is 14 upon 5 and y is minus 11 upon 5 You can check these values by putting them in the equation to see if your answer is correct or not Similarly you have 3 by 3 matrix, the process will be same First you have to make A matrix, 1, 1, 1, 2, y and here there is no z So 2 1 and 0 1, 1, minus 4 X matrix has 3 variables X, Y, Z B matrix has 10, 13, 0 The first thing to do is to get the determinant 1 We will leave this and this So minus 4 and 0 Means minus 4 minus 1, minus 8 and minus 0 1, minus 1 so this is minus 4 plus 8 plus 1 so 9 minus 4 is 5 so this is not equal to 0 so we can find the inverse to get the inverse we need adjoint and to get the adjoint we need cofactor we have taken cofactor in the back first we took minor then cofactor you can take cofactor directly because the method is same just change the sign and one for this column so all these three columns are cofactors of row 1 so here you have this was minus 4 write it here this was minus 8 change the sign and write it here this is 1 write 1 here here is 1 nothing here now here is second row first column second row first column so minus 4 minus 1 minus 5 will be written here write plus 5 here 2nd row 2nd column minus 4 minus 1 minus 5 2nd row 3rd column 1 minus 1 0 3rd row 3rd row 1st column 0 minus 1 3rd row 2nd column 0 minus 2 here we will change the sign plus 2 3rd row 3rd column 1 minus 2 here it will be minus so we have its adjoint transpose of co-factor minus 4, 8, 1, 5 minus 5, 0, minus 1, 2, minus 1 now take out inverse with help of this determinant was 5, divide by this add joint in INTO Now we have to find the value of x, so x will be equal to a inverse into b 1 upon 5 minus 4, 5 minus 1 8-5 to 10-1 into 10, 13, 0 Solve it We will do minus 4 from 10 to minus 40 13, 5, 65 Means 25 And this will be 0 at the last one So here 25 is there Then look at 10, 8, 80 13 x 5 is 65 minus 65 from 80 15 will come from here 1 x 10 is 10 this is 0 and this is 0 so this is 10 we can divide all these by 5 here 5, here 3, here 2 so this means value of x is 5 value of y is 3 and value of z is 2 The second rule is the Scrammer rule In this Scrammer rule, if you don't have D equal to 0 then we will say that the system is consistent and independent and it will have a unique solution How will we find the solution? D1 upon D and the value of Y will be D2 upon D How will we find D1 and D2? We will understand in the next question If D value is 0, then we have to check D1 and D2 if any one of these is non zero then we will say that the system is inconsistent and there is no solution for it but if we have all the zeroes like d, d1, d2 then we will say that the system is consistent, dependent and has infinite many solutions if you have equation 3 by 3 then the first condition will be like this if d0 equals to zero then the system is consistent and independent and there will be a unique solution x equals to y equals to d1 upon d y value is d2 upon d and z value is d3 upon d but if d value is 0 then we will check if any of d1, d2, d3 is non-zero then we will say that the system is inconsistent and there is no solution for it but if all are zero then we will say that the system may or may not be consistent it is not necessary that it is consistent or inconsistent so we will have to check it In case the system is consistent then it will have an infinite number of solutions If it is consistent then it will have infinite solutions and the system will be dependent So how will we check it? We will convert the 3x3 equation into 2x2 and then we will do it Now here you have to check and find the solution To find the solution, first you have to find d So how will we find d? The coefficient of d is 2-3 5 and 3 check this, 3 is 6 5 is 15 21 is 0 this means the solution is possible now we will take out D1 how to take out D1 take the 5 and 2 in the first column keep the second column same 15 plus 6 21 is done now D2 D2 will be taken to the second column of column 502 The first column will remain the same 4-25 which is minus 21 If we calculate the value of x, it will be d1 upon d which is 21 upon 21 If we calculate the value of y, it will be d2 upon d which is minus 21 upon 21 which is minus 1 Here you have three equations. First you write D. 1, minus 2, 3, 2, 1, minus 1, 3, minus 1, 2 Check here 1. If we leave this and this, then see 2 and this minus 1. 1 is here. Then plus 2 will be written by changing the sign. If we leave this and this, 4 plus 3, 7 is here. 3 leave this and this, minus 2 and minus 3 minus 5 so you will get 1, 7, 2, 14 minus 15, means this is 0 if D0 is there then we will remove D1, D2, D3 if any non-zero is there then we will write no solution so first remove D1 in D1, in the right hand side the constant is 1, 3 and minus 2, we will write 1, 3 and minus 2 we will keep the rest safe now let's solve this 1, 2, minus 1. 1 is done. Sign change. We will take this. 3 to the 6, minus 2. 4, plus 3. We will subtract this. Minus 3, plus 2. It will be minus 1. 1, 4 to the 8, minus 3. 9, minus 3, 6. this is not equal to zero so when A is not equal to zero then there is no need to find D2 or D3 we will write no solution in the next question it says that three shopkeepers A, B, C are using polythene bag, handmade bag newspaper bag A uses 20, 30 and 40 number of bag of respectively of type and B uses 30, 40 and 20 and C uses 40, 20 and 30 Now each shopkeeper will spend Rs. 250, 220 and 200 on the bag Find the cost of each carry bag using matrix method So we will assume that let the cost of Polythene bag be Rs. X Handmade bag be Rs. Y and Newspaper bag be Rs. Z You have to write this line in the answer Now the equation will be How much is shopkeeper A using? 20 bags are being used by polythene so 20x cost is created then 30 and after that 40 and the cost of AS shopkeeper is 250 so 10 comma is coming out from here so the equation will be 2x 3y plus 4z equals to 25 this is the equation we have made first the second shopkeeper is using 30 40 20 30, 40 and 20 and its cost is 220 so if you subtract 10 common from here it will be 3x 4y 2z equals to 22 and the third shopkeeper is using 40, 20, 30 40, 20 and 30 The total cost of this equation is 200 From here 10 will be common So it will be 4x 2y 3z equals to 20 This equation will be third Okay Now make a of this equation a will be 234 342 423 Now we will calculate the determinant of this value Determinant of this value is minus 27 This is not equal to zero This means inverse of this value will exist Now we will calculate cofactors of this value C11, C12, C13, C21 C22, C23, C31, C32 and C33 So first row first column We will leave this 12 minus 4 8 is done First row second column 9 minus 8 1 is done but sign will change First row third column 6 minus 16 6-16 means minus 10 2nd row, 2nd row, 1st column, 9-8, 1 2nd row, 2nd column, 6-16, minus 10 We haven't changed the sign here 2nd row, 1st column, 9-8, 1, we will change the sign to minus 1 2nd row, 3rd column, 4, minus 12, minus 8 here it will be plus 8 3rd row first column 6 minus 16 so here minus 10 3rd row second column 4 minus 12 minus 8 here it will be plus 8 3rd row third column 8 minus 9 means minus 1 so adjoint we have 8 minus 1 minus 10 minus 1 minus 10, 8, minus 10, 8 and minus 1. This will be our adjoint. Okay. After that, then we have to find out its inverse so inverse is we will divide by the determinant 1 upon minus 27 into 8 minus 1 minus 10 minus 1 minus 10 8 minus 10 8 and minus 1 now we know that to find out the value of x we have to multiply the inverse by b so inverse is 1 upon minus 27, 8, minus 1, minus 10, minus 1, minus 10, 8, minus 10, 8, minus 1 into b b is 25, 22 and 20 we will simplify this so 25 is 200 200 minus 22 and minus 200 200 minus 200 is cancelled only minus 22 is left minus 25 minus 220 and 160 solve it and get minus 85 minus 250 and 176 and minus 20 minus 250 and minus 20 is equal to minus 270 minus 270 minus 176 is equal to minus 94 so we will get the value of x it will be minus 22 upon minus 27 which is equal to 22 upon 27 which is the cost of polythene bag we will get the value of y which is equal to minus 85 upon minus 27 85 upon 27 is the hand-meet bag and the cost of the newspaper bag is 94 upon 27 this is the answer next question is school plan to award 6000 in total to its student to reward for the certain value A school has decided to distribute 6,000 prizes to its students For that, they have set the criteria of honesty, regularity and hard work When 3 times the award money for hard work is added to the award money given for the honesty Let's do this first Let's assume the honesty prize is X And the regularity prize is Y Let's assume the hard work prize is Z This is X, this is Y and this is Z You have to specify this first Now the total prize money is 6000. So this is the equation. Now the second one is when 3 times the award money for hard work. Means Z. 3 times the money of the hard work is added to the award money given for honesty. For honesty we have X. Amount to rupees 11000. These two together. 11000 total is 6000 so let's check here 11000 is given so make the equation here first put x here y is not here so put 0y 3z equals to 11000 this is second then the award money for honesty means x and Hard Work and these two together is double the award money for regularity which is 2y if you add 2y here x-2y plus z equals to 0 this equation is now third now from these three equations you can subtract a,b,c,y,c,1,1,1 here we will get 1, 0, 3, 1, minus 2, 1 we will get its determinant its determinant is 6 then we will get its co-factor first row first column zero plus six 1st row 2nd column 1, minus 3, minus 2, sign change will happen. 1st row 3rd column, minus 2. 2nd row 1st column 1, plus 2, 3, minus 3 will happen. 2nd row 2nd column 1, minus 1. 2nd row 3rd column, minus 2, minus 1, minus 3, here it will be plus 3. 3rd row 1st column, 3. 3rd row 2nd column 3, minus 1, 2, here it will be minus 2. third row third column 0 minus 1 means minus 1 adjoint is 6 2 minus 2 minus 3 0 3 3 minus 2 minus 1 and inverse is 1 upon 6 6 minus 3 3 2 0 minus 2 minus 2, 3 and minus 1 this is inverse now we will do the same x equals to a inverse b we will put the value of a inverse here 6 minus 3 3 2, 0, minus 2 minus 2, 3, minus 1 into b b is 6000, 11000 and 0 so now see here 6x is 36 and here is 33 so if we subtract 33 from 36 then we get 3 after that see here 12 this is 0 this is 0 from here subtract 12 and 33 subtract 12 from 33 so from 33 you will subtract 12 and get 21 now divide 6 into 5 500 is 2000 and here 6 is 18 and 6 is 30 so the price money is 500 Rs. 2000 for this Rs. 3500 for the third one These are all their prize money With this question, we will finish the Determined one shot I hope you have understood the concept of Determined You must have done the revision well And we hope you will not leave any question in the board exam That's all for today's video Thanks to all of you

Now after that, let's remove the minor. So what is a minor? A minor is also a determinant of order n.

n should be greater than or equal to 2. The value of n should be greater than or equal to 2. That means a minor of 1 by 1 should not be taken. To take a minor, at least a 2 by 2 order matrix should be there. Now how will we take a minor? Then the determinant of order n minus 1 obtained from the determinant after deleting its ith row and jth column is called the minor of the element. So, it has become a bit theoretical, let's see how to do it in a bit practical way.

Suppose you have A matrix. 1, 2, 3, 4. What you have to do is to find the minor. So, first you have to find the minor of this element.

So, we denote the minor as m11. That is, find the minor of the first row and first column. So, it is saying that the row and column of the element you want to find the minor of, delete it.

That is, this row. and this column is deleted here you have 4 left this row and this column are deleted 4 left, so this is your answer this is how you will find the minor first row second column first row second column this delete you have 3 left second row first column second row first column you have 2 left second row second column you have 1 left so these are their minors Now we have to find the cofactor. The value of the minor you have found will remain the same. We denote the cofactor with capital A or C.

I will write it with capital C. The value of the cofactor will remain the same. The only difference is the sum of i and j is odd.

ij means the number written in the base as you can see here 1 and 2 have odd sum so you have to change the sign if it is plus then you have to minus and if it is minus then you have to plus as you can see here the sum is also odd so you have to change the sign this is the difference so whether you are taking the minus of 2 by 2 or 3 by 3 the co-factor you will take in all of them, you have to change the sign of them where the sum of i and j is odd number will come now we will take out 3x3 cofactors and we will do that in the next questions here I took 2x2 to explain the concept now it is said that we have already done this that if you take out any determinant then you expand any row or column so that your determinant remains the same now you have two determinants so now we have determined the value of x so here it is not like a matrix that we have taken the corresponding element and taken x as 4 equals to 6 so we have got the value of x as 4 now we have to find the determinant 3 into 0 is 0 then minus sign minus 24 from here we have minus 3 then minus sign and here we have x cube so this is plus 24 take this minus 3 here and it will be 27 you can write 27 as minus 3 cube and write it as minus x cube if you compare both then you will get x value x value will be 3 minus 3 should be there here it is plus 27 so we write it as this and we write it as this so we get x value minus 3 ok here it says find the minor of the element 7 in the determinant this now here 3 by 3 you have it is asking for the minor of element 7 where is element 7? here it is it is asking for its minor so which place is it? 2nd row 3rd column so 2nd row 3rd column minor means delete 2nd row and delete 3rd column now the remaining you have find its determinant minus 4 then minus sign 3 into 1 is 3. Then it will be minus 7. Now if you were asked about the cofactor of this, if you were asked about the cofactor of this, then we would write 7. Why? Because look at its sum.

2 over 3. 5 is its sum. Odd is its sum. So whatever sign had to come here, you have to reverse that sign.

If it is minus, then you have to do plus. If it is plus, then you have to do minus. Find the minor and cofactor of A and check whether this. So here you have to remove all the minor and cofactors So we have said to remove the minor M11 M12, M13 then M21, M22, M23 M31, M32, M33 Solve this M11 first row, first column will be left determine this, 4 will be 0 then minus sign in the middle pi 4 is 20 so this is minus 20 now first row we will leave second column 6 7 is 42 then minus sign 4 1 is 4 so this is minus 46 first row third column 30 minus 0 so this will be 30 after that come to second row 2nd row 1st column so 2nd row and 1st column means minus 3 and minus 7 21 5 5 goes to 25 means it becomes minus 4 2nd row 2nd column minus 14 and minus 5 2nd row 3rd column 10 from here it is minus 3 so it becomes plus 13 third row first column third row first column minus 12 minus 5 into 0 third row second column third row second column 8 minus the third row second column 8 minus 30 so I'll take a minus 22 third row third row third column 0 minus minus 18 so this is plus 18 so this is the minus now you can calculate the co-factor what will be the co-factor let's denote it C11, C12, C13 C21, C22 C23 C31 C32 and C33 So C11 means You have this value of minus 20 Which sign you have to change in cofactor Which has some odd So what is the sum odd See this is the sum odd This is the sum odd This is the sum odd This is the sum odd You have to change the sign of these values Here minus 46 is coming Here we will write plus 46 Here 30 is coming here minus 4 is coming, plus 4 here minus 19 so this will remain the same here 13 minus 13 here minus 12 this will be plus 12 here minus 22 this will remain the same, minus we have to change the sign, so this will be plus 22 this will be 18 Now it is saying that the cofactor is gone Now it is asking us to verify this property So this property says that if we take any row element And if we multiply it with any other cofactor Then the answer will always be 0 If we multiply the row we are talking about with the same cofactor Then the answer always comes equal to the determinant But if we take any other row then the answer will always be 0 See how we are verifying here a is a cofactor we have written c here and given a there so here a11 a11 means 2 into a31 a31 means c31 c31 is minus 12 plus a12 first row second column means minus 3 into a32 32 is 22 plus a13 which is 5 into a33 which is 18 so from here it will be minus 24 minus 66 plus 90 if we add these two minus 90 plus 90 the answer is 0 so we have verified this property Next, you need to remember the properties of determinant These are the properties that will help you solve the questions These properties will be useful in MCQ questions Here, you don't have to do the questions like prove that You will use these properties in the rest of the questions In the first property, if each element in a row or column of a determinant is 0 Then the value of determinant is 0 In any determinant, if you want to solve the question All the elements are given zero in row or column So write anything here It doesn't matter anything, the answer will always be zero If each element on one side of the principal diagonal of a determinant is zero Then the value of determinant is the product of the diagonal element This means that if you have a diagonal element on one side of the diagonal element If the elements on one side of it are given zero The determinant of this will be the answer of the diagonal element The multiplication of these 6 Now it can be 0 here or here It can be 1,2,3 So 0 will be here And something else here Then also the answer will be 0 And both sides will be 0 sorry, the answer is 6 the answer is 6 the diagonal element will be multiplied whether it is this situation or that situation the determinant will remain the same the value of determinant remains unchanged if its row and column are interchanged if you interchange row and column then there will be no difference in the determinant interchanged row and column means that you are transposing it so it is saying that if you are going to use matrix or the transpose of the determinant, then it will remain the same. This property means this.

If any two row or column of a determinant are interchanged, then the value of determinant changes by minus sign only. In the previous property, we were changing the row into column. We were making the first row into first column, i.e. we were doing transpose.

But here it is saying that if you have a matrix, you are taking out the determinant a, b, c. D E F G H I Let's assume that we have a delta here Now I will interchange one row with another row I will make the first row as the second row and the second row as the first row So the answer will be minus sign So it will be minus delta If each element of any row or column of a determinant are identical or proportional, then the value of determinant is zero. If any determinant is 2a, 2b, 2c, then the answer is zero. Because these two rows are proportional. So either it is proportional or identical.

so if here is a, b, c then also the answer will be zero and here if 2a, 2b, 2c means if you find the ratio of these two then also the answer will be zero if each element of any row or column of a determinant is multiplied by the same number k then the value of the new determinant is k times the value of the original determinant so here it says that if you have this determinant The answer is delta. If I multiply one row by K, K A, K B and K C, the answer is K multiplied in the determinant. If we multiply one row, it will be multiplied once.

If you multiply the second row, then k will be square if you multiply third row also then k will be cube so from here a property is created if you calculate the determinant of k into a k into a means we have multiplied all the elements with k and the order of this will be n by n then k will be multiplied here k will be multiplied by 3 times in the 3x3 matrix k will be multiplied by 3 times in the original determinant so this property will be multiplied by the order of k Next, the sum of the product of any row or column with the cofactor of the corresponding element of the sum other row for the column is 0 Here it is said that if you multiply any row or column element with its cofactor If you multiply by the cofactor of another row, the answer will always be 0 which we just proved We took the element of first row and the cofactor of third row We multiplied them and added them together So the answer is 0 So this is the same property If A and B are square matrices of the same order Then determinant of AB equals to determinant of A into determinant of B If you determine AB, you can write it like this Determinant of A into Determinant of b Now using this property we get a result That if you want to get the determinant of a to the power n Then you simply have to Get the determinant and do it n times So what will be the benefit of this? Suppose you want to get the determinant of a to the power 4 So you don't have to Multiply a by 4 You will get the determinant of a And do it to the power 4 So this will be easier Now the question is, a and b are square matrices of the order 3 such that determinant of a equals to minus 1, determinant of b equals to 3. Then find the value of this. So the properties we have done now, you have to use those properties.

First of all, take out the scalar. Its order is 3 by 3. So in this, we are using the property of k into a determinant. So how many times k comes out of this is the order of this So here you are taking out 3 How many times will you take out?

3 times So 3 power 3 into determinant of AB Now here you put the property that determinant of AB is equal to this So 27 into determinant of A is minus 1 and this is 3 So it will be minus 81 If A is a square matrix and determinant of A equals to 2, then write the value of this when A is the transpose of matrix A. You have to find its determinant. So, you can apply this property here that we will do determinant of A and determinant of A transpose separately. So, the result remains the same.

Now, we have also read a property that whether you find A's determinant or its transpose's determinant, both are equal. So, I can write the determinant of A here So, this will be the square of the determinant of A A is the determinant of 2 2 power 2 means the answer is 4 Now, the concept is Area of Triangle Area of Triangle is the area of the three vertex of a triangle The formula of area of triangle is The sign here is the sign of determinant not of matrix Determinant you have to find is x1,y1,x2,y2,x3,y3 Keep this and in the last column always keep 1,1,1 If you solve this then this will be your area of triangle If your area of triangle is negative because area is never negative then we can neglect the minus sign If you find the area of triangle is minus 10, then the final answer is area of triangle equals to 10 square units. If you find the area of triangle is zero, then the points are collinear points.

Because when the points are in the same line, then the triangle will never be formed. If the triangle is not formed, then the area of triangle will not be formed. So the area of triangle will be equal to zero.

Now let's find the area of triangle So what was the formula? 1 by 2 In the formula write 5, 4, 2, minus 6, minus 2, 4, 1, 1, 1 Simplify this First we took 5, minus 6 and from here minus 4 then we will change the sign this one and this one this is minus 2 and the sign in between is minus plus 2 this one and this one 8 minus 12 this will be minus 10 10, 5 is 50 4 is 16 minus 4 So, everything is in minus, so see how much is here, here 20, 50, 70 is here. So, 1 into minus 70 means minus 35. So, when you write the area of this, you will not write minus 35. You will write the area of triangle is 35 square unit. For what value of k the point are collinear?

Now tell me the value of k to make this point collinear Collinear means that the area of triangle is equal to zero because the point is collinear If you take the area of triangle equal to zero, then it will be half From here, 3 minus 2, 8, 8, k, 2, 1, 1, 1 So this half will go to zero, its work is finished With 3, 8 minus 2 6 is done. Then we will write by changing its sign. We will leave this and this.

8 minus k plus 1. We will leave this and this. 16 minus 8k. From here 18 will come.

From here 2 is 16 minus 2k. From here 16 minus 8k. So minus 8 minus 2 minus 10k is done.

So, we will take 10k there. 16 is 32 and 18. 50 equals to 10k. So, k will be 5. After that, the next question is in the next concept, we have adjoint.

What does adjoint mean? Let a be the square matrix of order greater than or equal to 2. Means, the minimum order is 2. then adjoint of A is the transpose of cofactor if we write the cofactor transpose then it becomes adjoint of A and we will denote it as adjoint of A it has some properties of adjoint if you write adjoint of A then we write reverse law in transpose similarly you will write reverse law here adjoint of B adjoint of A To find the adjoint of A transpose, you can find the adjoint of A and then transpose it. Adjoint of adjoint of A.

So, what will be its formula? Determinant of A power n minus 2 into A. Here, the determinant of adjoint of A. Its formula is determinant of A power n minus 1. If you add A with it, then it will be determinant of A power n.

a into adjoint of a or adjoint of a into a so this will be equal to determinant of a into i so keep these properties in mind and use them in questions then comes singular matrix you can call a matrix singular only when determinant of a is 0 if determinant of a is not 0 then we will call it non singular matrix now it is saying that you have to tell the value of k if it is a singular matrix So, singular matrix means that its determinant is 0. So, we will get its determinant as minus 8 minus 3k equals to 0. So, 3k's value is minus 8. That means k's value will be minus 8 upon 3. Inverse of a matrix. A square matrix A of order n is invertible of their existing square matrix of the same order such that AB equals to BA equals to I. This means that if we do AB, or BA, which is equal to I In that case, we will say that A inverse is equal to B You can say that B is the inverse of A and you can also say that A is the inverse of B if AB and BA are equal to identity matrix Every invertible matrix possesses a unique inverse Whichever invertible matrix exists, you can find its inverse then A will be the inverse of A. It will not happen that two different inverses of one matrix will come.

If A is an invertible matrix then inverse of A is this. If you want to get inverse of A, then it will be equal to this. We will denote inverse of A as such. A square matrix is invertible if and only if it is non-singular. If matrix is non-singular, then only its inverse will be derived, otherwise it will not be derived.

If determinant of A is 0, then its inverse will not be derived. Because the formula to get inverse is adjoint of A divided by determinant of A. So it is not possible to get inverse. That is why determinant of A should always be non-zero.

Means non-singular matrix should be formed. If A and B are two invertible matrix of the same order then A B is also invertible. If A and B are both invertible matrix, means we can get both inverse. So the matrix which will come even after multiplying A B will be invertible. And the inverse of A B will be equal to B inverse into A inverse.

So here also reverse law is applied. So reverse law is applied in transpose also. It is applied in the adjoint and in the inverse as well. If A is an invertible matrix, then A transpose is also invertible.

If A inverse is obtained, then A transpose inverse will also be obtained. And if you do inverse of A transpose or inverse of transpose, then both of them remain equal. If A is an invertible matrix, then determinant of A inverse.

If you want to find the determinant of A inverse, then simply 1 upon determinant of A will be obtained. For what value of k the matrix has no inverse? He says that the matrix does not have an inverse. Inverse does not mean that the matrix is singular.

If it has no inverse, then determinant of a equals to 0. So, we will get the determinant of 10 minus 3k equals to 0. So, the value of 3k is 10 and the value of k is 10 by 3. Okay. If the matrix is in non-singular, then find the value of x. If the matrix is non-singular, then non-singular means that the determinant of A is not equal to 0. So, find the determinant of A.

6, minus 2 and minus 10. Minus 12, minus x. We will leave this column as 4 plus 20 and this is 24 and we will leave this column as 10 minus 10 which is 0 0 is not equal to 0 So, this will be minus 72 and this will be minus 24x plus 0 is not equal to 0 Take 72 there So, the value of x will be not equal to 72 upon minus 24. That means, x should not be equal to minus 3. If a equals to this, then find x such that this. Then find a inverse. This means that, first of all, if you have this a, then you have to find the value of x, if we have this relation. So, according to this relation, a square xA is equal to xA-2i a square means 3-2, 4-2 equals to xA-2i if we want to square it, we have to write it again so this will be 9 minus 8, 1 is done.

minus 6 plus 4, minus 2, 12 minus 8, 4, minus 8 plus 4, from here 3x minus 2x, 4x minus 2x, even get 2, 0, 0, 2. 3x-2, minus 2x, and 4x, minus 2x, plus 2. So, from here, since both the matrices are equal, keep this element equal to this. 3x-2 equals to 1. So, 3x equals to 3. So, the value of x is 1. But, is the value of x coming from each one or not? Do check this. So if you put 1 here, it will become minus 2. So here minus 2 is coming.

If you put it here, it will become 4. 4 is coming. If you put it here, it will become... It will be zero but here it is minus 4. So we have done something wrong here.

Okay, here it is minus 2. So here it will be minus 2. Now if you keep it here, it will be minus 4. So always check that if the value of x is different from two places, then it means that there is no value of x. So, we have got the value of x. The value of x is 1. Now, it is saying that hence find A inverse.

Now, find its inverse as well. Now, how will we find the inverse? Whenever such a question arises, we will not directly apply the inverse formula that adjoint of A divided by determinant. We will use this equation. Now, we have got A square equals to x into A minus 2 into I.

So, what we do is we multiply A inverse here as well. we multiply here as well and here as well if we multiply A inverse with this, then A will be saved if we multiply A with A inverse, then identity is saved and if we multiply identity with any matrix, then the same matrix is saved now inverse is here, take it this side and take it this side We have taken the value of x as 1. So, we will put 1 here. Let's put 1 first.

Now, we will put 1. If we put 1 instead of x, then we will have i. We will take a here. Let me show you because x equals to 1. So, you have i is 1, 0, 0, 1 a is 3 minus 2 4 minus 2 So we have to find A inverse So we will take this 2 here 1 by 2 1 minus 3 minus 2 0 plus 2 0 minus 4 1 plus 2 Divide all these by 2 So minus 1, 1 minus 2 and 3 by 2 This will be your answer After this we have solution of a system of linear equation using inverse of matrix i.e. matrix method. There are chances of getting question of matrix method. In matrix method, we do solution and we just have to tell if the system is consistent or not.

So how do we check consistency? If you have a non singular matrix, i.e. if we have determinant of A is not equal to zero then we will say that the unique solution of the system is and how to find the solution? x equals to a inverse b we will use this to solve the question and system is called consistent and we will say that system is consistent we can find a solution to this but if we have a singular matrix which means if determinant of a is equal to 0 then you have to calculate b in adjoint of a into if b in adjoint of a into is equal to 0 then we will say that system is consistent but there is no solution to it, there are infinite solutions to it but if b is not equal to 0 in adjoint of a into then we will say that there is no solution to the system and its solution is inconsistent, so its solution is not possible here it is telling you to solve it with inverse of coefficient matrix So here we have to use the help of the equation to make the matrix A So here we will take the coefficient of x and y Then we will make a matrix B We will write x here So in x we will take the variable we are using So we are using x and y You have to write this in the column And write the constant in the right hand side with B so we have 5 and minus 1 now the first thing we have to do is to find the determinant of A because if it is 0, then we have to check the solution whether it is possible or not so here we have done, it will be 3 then minus 2, which means the sign in the middle is minus, plus 2 which means 5 which means 0 is not equal to 0 0 is not equal to 0 means we can find the inverse of it To find the inverse, we need to find its adjoint We used to transpose the cofactor But here we have a 2x2 matrix If we find the adjoint of 2x2 matrix, we can find its direct In the 2x2 matrix, we need to change these two elements So, instead of 1, 3, we have 3, 1. And we have to change the sign of both of them.

This will be 1 and this will be minus 2. So, you can always directly remove the adjoint of 2 by 2. But to remove 3 by 3, you have to remove its cofactor. Now, you have to remove its inverse from here. The inverse formula is adjoint of A divided by determinant of A. Now, what is the adjoint you have?

3, 1, minus 2, 1. we have to divide it by its determinant which is 5 so we will write 1 upon 5 this is the inverse now the relation we have made according to this we have if we multiply a by x then it is equal to b so we have to find the value of x so we will take a there which means a inverse into b now the value of a inverse is 1 upon 5 3x1-2x1 into b b is 5 minus 1 we will solve this 3 x 5 is 15 minus 3 15 minus 3 is 12 no 15 minus 1 is 14 this will be minus 10 and minus 1 minus 11 we will divide by 5 this will be 14 upon 5 minus 11 upon 5 So you will get the value of x is 14 upon 5 and y is minus 11 upon 5 You can check these values by putting them in the equation to see if your answer is correct or not Similarly you have 3 by 3 matrix, the process will be same First you have to make A matrix, 1, 1, 1, 2, y and here there is no z So 2 1 and 0 1, 1, minus 4 X matrix has 3 variables X, Y, Z B matrix has 10, 13, 0 The first thing to do is to get the determinant 1 We will leave this and this So minus 4 and 0 Means minus 4 minus 1, minus 8 and minus 0 1, minus 1 so this is minus 4 plus 8 plus 1 so 9 minus 4 is 5 so this is not equal to 0 so we can find the inverse to get the inverse we need adjoint and to get the adjoint we need cofactor we have taken cofactor in the back first we took minor then cofactor you can take cofactor directly because the method is same just change the sign and one for this column so all these three columns are cofactors of row 1 so here you have this was minus 4 write it here this was minus 8 change the sign and write it here this is 1 write 1 here here is 1 nothing here now here is second row first column second row first column so minus 4 minus 1 minus 5 will be written here write plus 5 here 2nd row 2nd column minus 4 minus 1 minus 5 2nd row 3rd column 1 minus 1 0 3rd row 3rd row 1st column 0 minus 1 3rd row 2nd column 0 minus 2 here we will change the sign plus 2 3rd row 3rd column 1 minus 2 here it will be minus so we have its adjoint transpose of co-factor minus 4, 8, 1, 5 minus 5, 0, minus 1, 2, minus 1 now take out inverse with help of this determinant was 5, divide by this add joint in INTO Now we have to find the value of x, so x will be equal to a inverse into b 1 upon 5 minus 4, 5 minus 1 8-5 to 10-1 into 10, 13, 0 Solve it We will do minus 4 from 10 to minus 40 13, 5, 65 Means 25 And this will be 0 at the last one So here 25 is there Then look at 10, 8, 80 13 x 5 is 65 minus 65 from 80 15 will come from here 1 x 10 is 10 this is 0 and this is 0 so this is 10 we can divide all these by 5 here 5, here 3, here 2 so this means value of x is 5 value of y is 3 and value of z is 2 The second rule is the Scrammer rule In this Scrammer rule, if you don't have D equal to 0 then we will say that the system is consistent and independent and it will have a unique solution How will we find the solution? D1 upon D and the value of Y will be D2 upon D How will we find D1 and D2? We will understand in the next question If D value is 0, then we have to check D1 and D2 if any one of these is non zero then we will say that the system is inconsistent and there is no solution for it but if we have all the zeroes like d, d1, d2 then we will say that the system is consistent, dependent and has infinite many solutions if you have equation 3 by 3 then the first condition will be like this if d0 equals to zero then the system is consistent and independent and there will be a unique solution x equals to y equals to d1 upon d y value is d2 upon d and z value is d3 upon d but if d value is 0 then we will check if any of d1, d2, d3 is non-zero then we will say that the system is inconsistent and there is no solution for it but if all are zero then we will say that the system may or may not be consistent it is not necessary that it is consistent or inconsistent so we will have to check it In case the system is consistent then it will have an infinite number of solutions If it is consistent then it will have infinite solutions and the system will be dependent So how will we check it?

We will convert the 3x3 equation into 2x2 and then we will do it Now here you have to check and find the solution To find the solution, first you have to find d So how will we find d? The coefficient of d is 2-3 5 and 3 check this, 3 is 6 5 is 15 21 is 0 this means the solution is possible now we will take out D1 how to take out D1 take the 5 and 2 in the first column keep the second column same 15 plus 6 21 is done now D2 D2 will be taken to the second column of column 502 The first column will remain the same 4-25 which is minus 21 If we calculate the value of x, it will be d1 upon d which is 21 upon 21 If we calculate the value of y, it will be d2 upon d which is minus 21 upon 21 which is minus 1 Here you have three equations. First you write D.

1, minus 2, 3, 2, 1, minus 1, 3, minus 1, 2 Check here 1. If we leave this and this, then see 2 and this minus 1. 1 is here. Then plus 2 will be written by changing the sign. If we leave this and this, 4 plus 3, 7 is here.

3 leave this and this, minus 2 and minus 3 minus 5 so you will get 1, 7, 2, 14 minus 15, means this is 0 if D0 is there then we will remove D1, D2, D3 if any non-zero is there then we will write no solution so first remove D1 in D1, in the right hand side the constant is 1, 3 and minus 2, we will write 1, 3 and minus 2 we will keep the rest safe now let's solve this 1, 2, minus 1. 1 is done. Sign change. We will take this. 3 to the 6, minus 2. 4, plus 3. We will subtract this.

Minus 3, plus 2. It will be minus 1. 1, 4 to the 8, minus 3. 9, minus 3, 6. this is not equal to zero so when A is not equal to zero then there is no need to find D2 or D3 we will write no solution in the next question it says that three shopkeepers A, B, C are using polythene bag, handmade bag newspaper bag A uses 20, 30 and 40 number of bag of respectively of type and B uses 30, 40 and 20 and C uses 40, 20 and 30 Now each shopkeeper will spend Rs. 250, 220 and 200 on the bag Find the cost of each carry bag using matrix method So we will assume that let the cost of Polythene bag be Rs. X Handmade bag be Rs. Y and Newspaper bag be Rs. Z You have to write this line in the answer Now the equation will be How much is shopkeeper A using?

20 bags are being used by polythene so 20x cost is created then 30 and after that 40 and the cost of AS shopkeeper is 250 so 10 comma is coming out from here so the equation will be 2x 3y plus 4z equals to 25 this is the equation we have made first the second shopkeeper is using 30 40 20 30, 40 and 20 and its cost is 220 so if you subtract 10 common from here it will be 3x 4y 2z equals to 22 and the third shopkeeper is using 40, 20, 30 40, 20 and 30 The total cost of this equation is 200 From here 10 will be common So it will be 4x 2y 3z equals to 20 This equation will be third Okay Now make a of this equation a will be 234 342 423 Now we will calculate the determinant of this value Determinant of this value is minus 27 This is not equal to zero This means inverse of this value will exist Now we will calculate cofactors of this value C11, C12, C13, C21 C22, C23, C31, C32 and C33 So first row first column We will leave this 12 minus 4 8 is done First row second column 9 minus 8 1 is done but sign will change First row third column 6 minus 16 6-16 means minus 10 2nd row, 2nd row, 1st column, 9-8, 1 2nd row, 2nd column, 6-16, minus 10 We haven't changed the sign here 2nd row, 1st column, 9-8, 1, we will change the sign to minus 1 2nd row, 3rd column, 4, minus 12, minus 8 here it will be plus 8 3rd row first column 6 minus 16 so here minus 10 3rd row second column 4 minus 12 minus 8 here it will be plus 8 3rd row third column 8 minus 9 means minus 1 so adjoint we have 8 minus 1 minus 10 minus 1 minus 10, 8, minus 10, 8 and minus 1. This will be our adjoint. Okay. After that, then we have to find out its inverse so inverse is we will divide by the determinant 1 upon minus 27 into 8 minus 1 minus 10 minus 1 minus 10 8 minus 10 8 and minus 1 now we know that to find out the value of x we have to multiply the inverse by b so inverse is 1 upon minus 27, 8, minus 1, minus 10, minus 1, minus 10, 8, minus 10, 8, minus 1 into b b is 25, 22 and 20 we will simplify this so 25 is 200 200 minus 22 and minus 200 200 minus 200 is cancelled only minus 22 is left minus 25 minus 220 and 160 solve it and get minus 85 minus 250 and 176 and minus 20 minus 250 and minus 20 is equal to minus 270 minus 270 minus 176 is equal to minus 94 so we will get the value of x it will be minus 22 upon minus 27 which is equal to 22 upon 27 which is the cost of polythene bag we will get the value of y which is equal to minus 85 upon minus 27 85 upon 27 is the hand-meet bag and the cost of the newspaper bag is 94 upon 27 this is the answer next question is school plan to award 6000 in total to its student to reward for the certain value A school has decided to distribute 6,000 prizes to its students For that, they have set the criteria of honesty, regularity and hard work When 3 times the award money for hard work is added to the award money given for the honesty Let's do this first Let's assume the honesty prize is X And the regularity prize is Y Let's assume the hard work prize is Z This is X, this is Y and this is Z You have to specify this first Now the total prize money is 6000. So this is the equation. Now the second one is when 3 times the award money for hard work. Means Z.

3 times the money of the hard work is added to the award money given for honesty. For honesty we have X. Amount to rupees 11000. These two together.

11000 total is 6000 so let's check here 11000 is given so make the equation here first put x here y is not here so put 0y 3z equals to 11000 this is second then the award money for honesty means x and Hard Work and these two together is double the award money for regularity which is 2y if you add 2y here x-2y plus z equals to 0 this equation is now third now from these three equations you can subtract a,b,c,y,c,1,1,1 here we will get 1, 0, 3, 1, minus 2, 1 we will get its determinant its determinant is 6 then we will get its co-factor first row first column zero plus six 1st row 2nd column 1, minus 3, minus 2, sign change will happen. 1st row 3rd column, minus 2. 2nd row 1st column 1, plus 2, 3, minus 3 will happen. 2nd row 2nd column 1, minus 1. 2nd row 3rd column, minus 2, minus 1, minus 3, here it will be plus 3. 3rd row 1st column, 3. 3rd row 2nd column 3, minus 1, 2, here it will be minus 2. third row third column 0 minus 1 means minus 1 adjoint is 6 2 minus 2 minus 3 0 3 3 minus 2 minus 1 and inverse is 1 upon 6 6 minus 3 3 2 0 minus 2 minus 2, 3 and minus 1 this is inverse now we will do the same x equals to a inverse b we will put the value of a inverse here 6 minus 3 3 2, 0, minus 2 minus 2, 3, minus 1 into b b is 6000, 11000 and 0 so now see here 6x is 36 and here is 33 so if we subtract 33 from 36 then we get 3 after that see here 12 this is 0 this is 0 from here subtract 12 and 33 subtract 12 from 33 so from 33 you will subtract 12 and get 21 now divide 6 into 5 500 is 2000 and here 6 is 18 and 6 is 30 so the price money is 500 Rs. 2000 for this Rs. 3500 for the third one These are all their prize money With this question, we will finish the Determined one shot I hope you have understood the concept of Determined You must have done the revision well And we hope you will not leave any question in the board exam That's all for today's video Thanks to all of you

Transcript for:Understanding Determinants and Their Applications

Transcript for:
Understanding Determinants and Their Applications